QUANTIZED SYSTEMS AND CONTROL. Daniel Liberzon. DISC HS, June Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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1 QUANTIZED SYSTEMS AND CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign DISC HS, June 2003

2 HYBRID CONTROL Plant: u y P Classical continuous feedback paradigm: u y P C But logical decisions are often necessary: The closed-loop system is hybrid u y P l o g i c C 1 C 2

3 REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

4 REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

5 CONSTRAINED CONTROL 0 Control objectives: stabilize to 0 or to a desired set containing 0, exit D through a specified facet, etc. Constraint: given control commands

6 LIMITED INFORMATION SCENARIO partition of D points in D, Quantizer/encoder: for Control:

7 MOTIVATION Limited communication capacity many systems sharing network cable or wireless medium microsystems with many sensors/actuators on one chip Need to minimize information transmission (security) Event-driven actuators PWM amplifier manual car transmission stepping motor finite subset of Encoder Decoder QUANTIZER

8 QUANTIZED CONTROL ARCHITECTURES PLANT PLANT STATE QUANTIZER OUTPUT QUANTIZER CONTROLLER CONTROLLER PLANT PLANT INPUT QUANTIZER INPUT QUANTIZER OUTPUT QUANTIZER CONTROLLER CONTROLLER

9 QUANTIZER GEOMETRY is partitioned into quantization regions uniform logarithmic arbitrary Dynamics change at boundaries => hybrid closed-loop system Chattering on the boundaries is possible (sliding mode)

10 QUANTIZATION ERROR and RANGE Assume such that: is the range, is the quantization error bound For, the quantizer saturates

11 EXAMPLES of QUANTIZERS A/D conversion Temperature sensor normal too high too low Camera with zoom Tracking a golf ball Coding and decoding

12 OBSTRUCTION to STABILIZATION Assume: M, fixed Asymptotic stabilization is usually lost

13 BASIC QUESTIONS What can we say about a given quantized system? How can we design the best quantizer for stability? What can we do with very coarse quantization? What are the difficulties for nonlinear systems?

14 BASIC QUESTIONS What can we say about a given quantized system? How can we design the best quantizer for stability? What can we do with very coarse quantization? What are the difficulties for nonlinear systems?

15 STATE QUANTIZATION: LINEAR SYSTEMS is asymptotically stable 9 Lyapunov function Quantized control law: where is quantization error Closed-loop system:

16 LINEAR SYSTEMS (continued) Previous slide: Recall: Combine: Lemma: solutions that start in enter in finite time

17 NONLINEAR SYSTEMS For linear systems, we saw that if gives then automatically gives when This is robustness to measurement errors For nonlinear systems, GAS such robustness To have the same result, need to assume when This is input-to-state stability (ISS) for measurement errors!

18 SUMMARY: PERTURBATION APPROACH 1. Design ignoring constraint 2. View as approximation 3. Prove that this still solves the problem Issue: Need to be ISS w.r.t. measurement errors error

19 INPUT QUANTIZATION Control law: where Closed-loop system: Analysis same as before Control law: where Closed-loop system: Need ISS with respect to actuator errors

20 OUTPUT QUANTIZATION Control law: Closed-loop system: Analysis same as before (need a bound on initial state) Can also treat input and state/output quantization together

21 BASIC QUESTIONS What can we say about a given quantized system? How can we design the best quantizer for stability? What can we do with very coarse quantization? What are the difficulties for nonlinear systems?

22 LOCATIONAL OPTIMIZATION: NAIVE APPROACH Smaller => smaller for Also true for nonlinear systems ISS w.r.t. measurement errors This leads to the problem: Compare: mailboxes in a city, cellular base stations in a region

23 MULTICENTER PROBLEM Critical points of satisfy 1. is the Voronoi partition : 2. Each is the Chebyshev center (solution of the 1-center problem). This is the center of enclosing sphere of smallest radius Lloyd algorithm: iterate

24 LOCATIONAL OPTIMIZATION: REFINED APPROACH Revised problem: only need this ratio to be small Logarithmic quantization: Lower precision far away, higher precision close to Only applicable to linear systems

25 WEIGHTED MULTICENTER PROBLEM on not containing 0 (annulus) Critical points of satisfy 1. is the Voronoi partition as before 2. Each is the weighted center (solution of the weighted 1-center problem) This is the center of sphere enclosing with smallest Lloyd algorithm as before Gives 25% decrease in for 2-D example

26 DYNAMIC QUANTIZATION: IDEA Temperature sensor can adjust threshold settings Digital camera can zoom in and out Encoder can change the coding mechanism zoom out zoom in Zoom out to overcome saturation After ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability (also applies to input and output quantization)

27 DYNAMIC QUANTIZATION: DETAILS zooming variable Hybrid quantized control: is discrete state (More realistic, easier to design and analyze, robust to time delays) unknown Increase fast enough until We know: solutions starting in enter in finite time after units of time dwell time

28 BASIC QUESTIONS What can we say about a given quantized system? How can we design the best quantizer for stability? What can we do with very coarse quantization? What are the difficulties for nonlinear systems?

29 ACTIVE PROBING for INFORMATION PLANT QUANTIZER CONTROLLER dynamic (time-varying) dynamic (changes at sampling times) Encoder Decoder very small

30 LINEAR SYSTEMS Example: Zoom out to get initial bound sampling times Between sampling times, let

31 LINEAR SYSTEMS Example: Between sampling times, let Consider The norm grows at most by the factor in one period is divided by 3 at the sampling time

32 LINEAR SYSTEMS (continued) The norm grows at most by the factor in one period is divided by 3 at each sampling time Pick small enough s.t. sampling frequency vs. open-loop instability amount of static info provided by quantizer where is Hurwitz 0

33 Example: NONLINEAR SYSTEMS Zoom out to get initial bound sampling times Between samplings

34 Example: NONLINEAR SYSTEMS Between samplings Let where is Lipschitz constant of The norm grows at most by the factor in one period is divided by 3 at the sampling time

35 NONLINEAR SYSTEMS (continued) The norm grows at most by the factor in one period is divided by 3 at each sampling time Pick small enough s.t. Need ISS w.r.t. measurement errors!

36 RESEARCH DIRECTIONS Robust control design Locational optimization Performance Applications

37 REFERENCES Brockett & L, 2000 (IEEE TAC) Bullo & L, 2003 (submitted)